In mathematics, value may refer to several, strongly related notions. Though in general, a mathematical value is a broad term that refers to any definite entity that can be manipulated with operators according to the well-defined rules of its mathematical system.
Certain values can correspond to the real world, although most values in mathematics generally exists purely as abstract objects with no connection to the real world.
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Continuity is one of the core concepts of calculus and mathematical analysis, where arguments and values of functions are real and complex numbers. The concept has been generalized to functions between metric spaces and between topological spaces. The latter are the most general continuous functions, and their definition is the basis of topology.
Mathematical optimization (alternatively spelled optimisation) or mathematical programming is the selection of a best element, with regard to some criteria, from some set of available alternatives. It is generally divided into two subfields: discrete optimization and continuous optimization. Optimization problems arise in all quantitative disciplines from computer science and engineering to operations research and economics, and the development of solution methods has been of interest in mathematics for centuries.
In the more general approach, an optimization problem consists of maximizing or minimizing a real function by systematically choosing input values from within an allowed set and computing the value of the function. The generalization of optimization theory and techniques to other formulations constitutes a large area of applied mathematics.
In mathematics, a variable (from Latin variabilis 'changeable') is a symbol, typically a letter, that refers to an unspecified mathematical object. One says colloquially that the variable represents or denotes the object, and that any valid candidate for the object is the value of the variable. The values a variable can take are usually of the same kind, often numbers. More specifically, the values involved may form a set, such as the set of real numbers.
The object may not always exist, or it might be uncertain whether any valid candidate exists or not. For example, one could represent two integers by the variables p and q and require that the value of the square of p is twice the square of q, which in algebraic notation can be written p = 2 q. A definitive proof that this relationship is impossible to satisfy when p and q are restricted to integer numbers isn't obvious, but it has been known since ancient times and has had a big influence on mathematics ever since.
In mathematics, a limit is the value that a function (or sequence) approaches as the argument (or index) approaches some value. Limits of functions are essential to calculus and mathematical analysis, and are used to define continuity, derivatives, and integrals.The concept of a limit of a sequence is further generalized to the concept of a limit of a topological net, and is closely related to limit and direct limit in category theory.The limit inferior and limit superior provide generalizations of the concept of a limit which are particularly relevant when the limit at a point may not exist.
In a positional numeral system, the radix (pl.Β radices) or base is the number of unique digits, including the digit zero, used to represent numbers. For example, for the decimal system (the most common system in use today) the radix is ten, because it uses the ten digits from 0 through 9.
In any standard positional numeral system, a number is conventionally written as (x)y with x as the string of digits and y as its base. For base ten, the subscript is usually assumed and omitted (together with the enclosing parentheses), as it is the most common way to express value. For example, (100)10 is equivalent to 100 (the decimal system is implied in the latter) and represents the number one hundred, while (100)2 (in the binary system with base 2) represents the number four.
A blank cheque or blank check in the literal sense is a cheque that has no monetary value written in, but is already signed. In the figurative sense, it is used to describe a situation in which an agreement has been made that is open-ended or vague, and therefore subject to abuse, or in which a party is willing to consider any expense in the pursuance of their goals. The term carte blanche (borrowed from French; lit.β'white card') is used in a similar way.
In mathematics, the word constant conveys multiple meanings. As an adjective, it refers to non-variance (i.e. unchanging with respect to some other value); as a noun, it has two different meanings:
For example, a general quadratic function is commonly written as:
In chemistry and quantum mechanics, the spin quantum number is a quantum number (designated s) that describes the intrinsic angular momentum (or spin angular momentum, or simply spin) of an electron or other particle. It has the same value for all particles of the same type, such as s = β 1/2β for all electrons. It is an integer for all bosons, such as photons, and a half-odd-integer for all fermions, such as electrons and protons.
The component of the spin along a specified axis is given by the spin magnetic quantum number, conventionally written ms. The value of ms is the component of spin angular momentum, in units of the reduced Planck constant Δ§, parallel to a given direction (conventionally labelled the zβaxis). It can take values ranging from +s to βs in integer increments. For an electron, ms can be either β ++1/2β or β β+1/2β .